c.1200, from Old English conjunction oþþe "either, or," related to Old Frisian ieftha, Middle Dutch ofte, Old Norse eða, Old High German odar, German oder, Gothic aiþþau "or." This was extended in early Middle English (and Old High German) with an -r ending, perhaps by analogy with "choice between alternative" words that ended thus (e.g. either, whether), then reduced to oþþr, at first in unstressed situations (commonly thus in Northern and Midlands English by 1300), and finally reduced to or, though other survived in this sense until 16c.
The contraction took place in the second term of an alternative, such as either ... or, a common construction in Old English, where both words originally were oþþe (see nor).
word-forming element making nouns of quality, state, or condition, from Middle English -our, from Old French -our (Modern French -eur), from Latin -orem (nominative -or), a suffix added to pp. verbal stems. Also in some cases from Latin -atorem (nominative -ator).
In U.S., via Noah Webster, -or is nearly universal (but not in glamour, curious, generous), while in Britain -our is used in most cases (but with many exceptions: author, error, senator, ancestor, horror etc.). The -our form predominated after c.1300, but Mencken reports that the first three folios of Shakespeare's plays used both spellings indiscriminately and with equal frequency; only in the Fourth Folio of 1685 does -our become consistent.
A partial revival of -or on the Latin model took place from 16c. (governour began to lose its -u- 16c. and it was gone by 19c.), and also among phonetic spellers in both England and America (John Wesley wrote that -or was "a fashionable impropriety" in England in 1791).
Webster criticized the habit of deleting -u- in -our words in his first speller ("A Grammatical Institute of the English Language," commonly called the Blue-Black Speller) in 1783. His own deletion of the -u- began with the revision of 1804, and was enshrined in the influential "Comprehensive Dictionary of the English Language" (1806), which also established in the U.S. -ic for British -ick and -er for -re, along with many other attempts at reformed spelling which never caught on (e.g. masheen for machine). His attempt to justify them on the grounds of etymology and the custom of great writers does not hold up.
Fowler notes the British drop the -u- when forming adjectives ending in -orous (humorous) and derivatives in -ation and -ize, in which cases the Latin origin is respected (e.g. vaporize). When the Americans began to consistently spell it one way, however, the British reflexively hardened their insistence on the other. "The American abolition of -our in such words as honour and favour has probably retarded rather than quickened English progress in the same direction." [Fowler]